Carbon aerogels (CAs) are considered to be excellent candidate materials for supercapacitor applications, due to their high surface to volume ratio, chemical inertness, and low synthesis cost. However, performance of CA-based electrodes is limited at the nanoscale by low quantum capacitance, deriving from a small electronic density of states near the Fermi level. One strategy for overcoming these challenges is to tune the structure and chemistry of CAs at the nanoscale. However, this relies on a fundamental understanding of which structural features govern the electronic properties of the material, as well as how these might be modified for increased performance. To this end, we have performed first-principles calculations based on density functional theory to assess the effect of point defects, lattice strain, and local curvature on the quantum capacitance of single- and multilayer graphene. We find that certain features can have significantly affect the quantum capacitance, offering a design strategy for device improvement. Our results are combined with X-ray absorption spectroscopy in order to isolate the structural features present in laboratory samples, as well to deconvolve the total capacitance into contributions from the electrode and the electrolyte.